Characterizing of Inner Product Spaces by the Mapping $n_{x,y}$

Abstract

For the vectors $x$ and $y$ in a normed linear spaces $X$, the mapping $n_{x,y}:\mathbb{R}\to \mathbb{R}$ is defined by $n_{x,y}(t)=\Vert x+ty\Vert$. In this note, comparing the mappings $n_{x,y}$ and $n_{y,x}$ we obtain a simple and useful characterization of inner product spaces.